Hierarchical maximum likelihood estimation for time-resolved NMR data.

Metabolic monitoring and reaction rate estimation using hyperpolarized NMR technology requires accurate quantitative analysis of multidimensional data scenarios. Currently, this analysis is often performed in a two-stage procedure, which is prone to errors in uncertainty propagation and estimation. We propose an approach derived from a Bayesian hierarchical model that intrinsically propagates uncertainties and operates on the full data to maximize the precision at minimal uncertainty. In an analytic treatment, we reduce the estimation procedure to a least-squares optimization problem which can be understood as an extension of the Variable Projection (VarPro) approach for data scenarios with two predictors. We investigate the method's efficacy in two experiments with hyperpolarized metabolites recorded with conventional high-field NMR devices and a micronscale NMR setup using Nitrogen-Vacancy centers in diamond for detection, respectively. In both examples, the new approach improves estimates compared to Fourier methods and proves operational advantages over a two-stage procedure employing VarPro. While the approach presented is motivated by NMR analysis, it is straightforwardly applicable to further estimation scenarios with similar data structure, such as time-resolved photospectroscopy.